Loading the lesson
Loading the lesson
Search AllRounder.ai
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Learn it your way
A short class with Sarah and her students. The other ways are under More.
Sign up free to listen. Transcripts are free to read.
Today we'll classify partial differential equations by evaluating the discriminant b² - 4AC in the context of our equation 1 - M² φ_xx + φ_yy. Can anyone remind me what A, B, and C represent?
A is 1 - M², B is 0, and C is 1!
That's correct! Now, can anyone tell me what happens when M is less than 1?
The discriminant will be less than 0, indicating it's an elliptical PDE.
Right! And when M is greater than 1?
Then it becomes hyperbolic because the discriminant is greater than 0.
Excellent. Lastly, what about M equals 1?
It’s a parabolic PDE because the discriminant is exactly 0.
Exactly! Remember these classifications as they are crucial for understanding properties of PDEs. We will summarize these in a chart on the board.
Next, let's look at the function φ(x,y) = e^(x) + e^(y). We need to compute del φ/del x and del φ/del y at the point (1,1). What did we learn about computing partial derivatives?
We differentiate with respect to x and y separately.
Exactly! So can anyone compute del φ/del x?
It’s e^(x), so at (1,1) it equals e^(1), which is about 2.718.
Correct! Now for del φ/del y?
It's the same, e^(y) at (1,1) also gives 2.718.
Great! Next, we'll approximate these values using the finite difference method with delta x = delta y = 0.1. Who can set up the equation for del φ/del x at (i,j)?
It will be φ(i+1, j) - φ(i, j) divided by delta x!
Exactly! Now make sure to calculate the percentage difference from the exact value.
To recap, we differentiated φ to find the exact derivative and then used finite differences to approximate these calculations.
We have our values for the exact derivatives. Now let's compute the approximate values using our finite difference setup. The formula is del φ/del x at (i,j) = (φ(i+0.1,j) - φ(i,j)) / 0.1. What does the function evaluate to at i,j = 1,1?
For φ(1.1, 1), it will evaluate to e^(1.1) + e^(1), which is approximately 2.859.
Well done! How about del φ/del y?
Using the same method, it should be around 2.859 too.
Perfect! Finally, let’s calculate the percentage difference between our exact and approximate values.
The difference is calculated as (2.859 - 2.718) / 2.718 which gives us approximately 5.2%!
Great work! This exercise illustrates the concept of derivations and finite differences effectively, solidifying our understanding of φ(x,y).
The value derived from the coefficients A, B, and C that determines the nature of the PDE.
The categorization of a PDE into elliptical, parabolic, or hyperbolic depending on the discriminant.
A numerical technique used to estimate derivatives by approximating the function values at discrete points.
When M is less than one, an ellipse is done, when M goes over one, hyperbolics run, but if M’s one at all, parabolas stand tall.
There once was a mathematician who classified equations based on their discriminants. When he saw M less than one, he noted they shaped into ellipses. Once M exceeded one, they morphed into hyperbolas. And when M was exactly one, they proudly wore the title of parabolas.
Classify PDEs into E for Elliptic, H for Hyperbolic, and P for Parabolic. Remember: EHP!
M-CD: M is less (C) then Elliptic, M is greater (D) then Hyperbolic!
Term
What is a partial differential equation?
Click to flip
Term
What indicates an elliptical PDE?
Click to flip
Term
What is the finite difference method?
Click to flip
An equation that involves functions of multiple variables and their partial derivatives.
A type of PDE where the discriminant b² - 4AC < 0.
A type of PDE where the discriminant b² - 4AC > 0.
A type of PDE where the discriminant b² - 4AC = 0.
The derivative of a function with respect to one variable while holding other variables constant.
This section explores the solution to a problem involving the function φ(x,y) and discusses the classification of partial differential equations related to hyperbolic, parabolic, and elliptical types based on the parameter M.
The section details the calculation of the function φ and the interpretation of the discriminant b² - 4AC to classify the type of partial differential equations as elliptical, parabolic, or hyperbolic. It further includes an example of deriving derivative values using finite difference methods.
In this section, we investigate the function φ(x,y) through the framework of partial differential equations (PDEs). The essential equation stems from hydraulic engineering and is described as:
Classification of PDEs: We define A = 1 - M², B = 0, and C = 1 for the equation 1 - M² φ_xx + φ_yy. The discriminant b² - 4AC is calculated based on the parameter M. If M < 1, then the equation is classified as an elliptical PDE, whereas if M > 1, the behavior is hyperbolic. When M equals 1, the equation does not show hyperbolic or elliptical properties; instead, it is classified as parabolic.
Practical Example: The example presented involves computing the 1st derivatives of the function φ(x,y) = e^(x) + e^(y) at the point (1,1). Students are guided through deriving both exact values of the derivatives and their approximations using the first-order finite difference method with a specified delta value.
These calculations provide a practical understanding of differential equations in hydraulic engineering contexts, culminating in the analysis of the percentage difference between exact and approximate derivative values.